Isosoleces Triangle

By definition, an isosceles triangle is a triangle that has at least two congruent sides. This means that an equilateral triangle is an isosceles triangle.

Isosceles Triangle

Figure 1 Isosceles Triangle

\Delta{ABC} in figure 1 is an isosceles triangle because \overline{AB} \cong \overline{AC}. This would be by the definition of an isosceles triangle. There are other ways to justify a triangle is an isosceles triangle besides the definition. I talk about that in another post. I want to focus in on the parts of an isosceles triangle in this post.

For any isosceles triangle, the legs are congruent and form the vertex angle. The vertex angle is opposite the 3rd side of the triangle. The 3rd side is called the base and helps form the base angles. The base angles are opposite the legs and the base angles are congruent. For \Delta{ABC} in figure 1, \angle{A} is the vertex angle, \angle{B} \cong \angle{C} because they are base angles,  \overline{AB} \cong \overline{AC} because they are the legs and \overline{BC} is the base.


Find the Quadratic Function That Models 3 Ordered Pairs

Find the Quadratic Function That Models 3 Ordered Pairs

Figure 1 is the solution to the second example on a post about finding the equation to a parabola when given three points.  The directions from the previous post read, “Find the quadratic function that models the given ordered pairs: (-2,-17), (-1,10), (5,-10).”

I plan on writing an explanation, but for now I wanted to get the solution up!


Find the Quadratic Function That Models 3 Ordered Pairs

Figure 1

Modeling Data With Quadratic Functions

Find the Quadratic Function That Models 3 Ordered Pairs

In a previous post about quadratic functions and the graphs, I discussed the standard form of a quadratic equation, f(x)=ax^2+bx+c. Because a parabola is symmetric, its equation can be found by solving a system of equations with three variables. I have only discussed solving equations with 2 variables previously.  If you know how to solve a system with 2 variables, then a system with 3 variables should be no problem. The process is rather simple. I will explain how to solve the system of three equations as well as a system of two equations in the examples.

Example 1 Find the Quadratic Function That Models 3 Ordered Pairs

Before discussing how to find the solution, there should be a discussion about what is to be found. Ultimately, the solution should be an equation in the standard form of a quadratic, f(x)=ax^2+bx+c. To find this equation or function involves multiple parts and each part consists of several steps. Many times we get so caught up in the details, we forget the overall picture.

The answer is: f(x)=x^2-6x+7. Yes, I just gave you the answer. Now, the question is:

Find the quadratic function that models the given ordered pairs: (2,-1), (3,-2), (1,2). Here is how you turn 3 ordered pairs into a quadratic function.

First you must create a system of three equations. You create this system of equations by substituting the ordered pairs into the standard form of a quadratic equation f(x)=ax^2+bx+c. Using the reflexive and substitution properties of equality, I use the equation ax^2+bx+c=y. This will make solve the system of equations easier.

Substituting the ordered pairs gives the following equations:

(2,-1) gives a(2^2)+b(2)+c=-1

(3,-2) gives a(3^2)+b(3)+c=-2

(1,2) gives a(1^2)+b(1)+c=2

Simplifying these equations, maintaining the order from above and labeling them as equations A, B and C results with:

4a+2b+c=-1 (A)

9a+3b+c=-2 (B)

a+b+c=2 (C)

Now that you have a system of three equations, it is good to review the overall goal, to find the coefficients of a quadratic function that models the ordered pairs given. Okay, on with solving the system of 3 equations. You need to use the 3 equations and the elimination method to create a new system of equations with only 2 variables. You have to choose two pairs of equations and eliminate a variable. For this example, I will subtract C from A to create a new equation D and I will subtract C from B to create equation E.

(A) 4a+2b+c=-1

\underline{-(C) a+b+c=2}

(D) 3a+b=-3

(B) 9a+3b+c=-2

\underline{-(C) a+b+c=2}

(E) 8a+2b=-4

Equations D and E form a new system of equations.

3a+b=-3 (D)

8a+2b=-4 (E)

To solve this system, it would be easiest to subtract with E from equation D multiplied by 2.

2(3a+b=-3) (D)

\underline{-(8a+2b=-4)} (E)

6a+2b=-6 (D)

\underline{-(8a+2b=-4)} (E)

-2a=-2  Divide both sides by -2 and


You now have the coefficient of the quadratic term. Using a = 1, you substitute into either equation D or E. For this problem, I used equation D.



3(1)+b=-3  Substitute 1 for a.

3+b=-3  Simplify.

b=-6  Subtract 3 from each side.

This is the coefficient of the linear term. Now use both a = 1 and b = -6 to find the value of the constant. You must use one of the original equations A, B or C. I used equation C because it is very simple.

a=1 and b=-6

a+b+c=2 (C)

1+(-6)+c=2  Substitute 1 for a and -6 for b.

-5+c=2  Simplify.

c=7  Add 2 to both sides.

Finally, we have all of the coefficients a = 1, b = -6 and c = 7 and can write the quadratic function that models the given points.

a=1, b=-6, c=7


f(x)=x^2-6x+7 This is the solution.

But how do you know it is correct? You should check your work by substituting the values a = 1, b = -6 and c = 7 into one of the equations A, B or C. Since I already used C I will use equation A.

a=1, b=-6, c=7

4a+2b+c=-1 (A)

4(1)+2(-6)+7=-1  Substitute.

4+(-12)+7=-1  Simplify.

-8+7=-1  Simplify.

-1=-1  Simplify and it checks.

This is just one example of finding the equation of a quadratic function from three points. Also, because the three original equations started with all three variables, it was a more complicated type.

Example Find the Quadratic Function That Models 3 Ordered Pairs For You To Do

Find the quadratic function that models the given ordered pairs:

(-2,-17), (-1,10), (5,-10).

Click here for the solution.

Quadratic Functions and Their Graphs

Standard Form of a Quadratic Function


Where a\neq0.

ax^2--> Quadratic Term

bx--> Linear Term

c--> Constant Term

The restriction on the quadratic term, a\neq0., ensures that the function is indeed a quadratic function. Without the squared term, you are left with a linear function or a constant function.

A simple skill assessed in most high school algebra 2 classes is the ability to classify functions. If you are studying quadratics, then you have studied linear equations with their close relative the constant function. To classify a function, it is often necessary to simplify what is given in order to properly determine the type of function.

Example 1 Identifying Functions

Classify each equation as linear or quadratic and identify the quadratic, linear and constant terms.
y=(4x+2)(-3x-1) (1)

y=(4x)(-3x) + (4x)(-1) + 2(-3x) + 2(-1) (2)

y=-12x^2 + (-4x) + (-6x) + (-2) (3)

y=-12x^2-10x-2 (4)

In the steps above, line 1 represents the given function. Line 2 represents simplifying the equation with multiplication. You may have heard of the acronym FOIL, which stands for: First, Outer, Inner, Last. This is a short cut to multiplying to binomials. Anyway, line 3 is formed by actually performing the multiplication set up in line 2. The change from line 3 to 4 occurs by collecting or adding the like terms of -4x and -6x to give the linear term -10x. The quadratic term is -12x^2 and the constant term is -2.

Example 2 Understanding the Graph of Parabola

As with working with linear functions, the ability to interpret and analyze the graphs of basic quadratic functions is critical. It is good to note that the graph of a quadratic function is called a parabola. Figure 1 is the graph to the parabola, f(x)=x^2+8x+14. For now, the function in the previous sentence is irrelevant. The reason for this, the graph is given. I will discuss in an upcoming post how the actual function is related to the graph of the related parabola.

Firgure 1 Parabola

Figure 1 Parts of a Parabola

In figure 1, f(x)=x^2+8x+14 is graphed and has vertex of (-4,-2) and an axis of symmetry (AOS) of x=-4. Though I did not label the other two distinguishable points, you should be able to see, that the other points on parabola correspond to each other. This is because parabolas are symmetric, hence the axis of symmetry. Those points are equidistant from x=-4. Since this parabola opens upward, it has a minimum or bottom value. You get the value from the y-coordinate. The reason you use the y-coordinate is there is not bound on the Domain. Parabolas extend in the both directions along the x-axis. Because of the shape of parabola, there is a limit on the range. In this case, the range could be expressed as {y|y≥-4}.

The previous paragraph can be boiled down to identifying the AOS, vertex, points and corresponding points on the graph of a quadratic function or parabola.

Figure 2 Parts of a Parabola

Figure 2 Parts of a Parabola

Example 3 Understanding the Graph of Parabola For you to do.

For figure 2, identify the axis of symmetry, the vertex, the minimum or maximum value and explain why it is a min or a max value, identify both points and their mirror image about the AOS.

Click here for the solution.

Parabolas – Identifing Parts of a Parabola

Example 3 Understanding the Graph of Parabola For you to do.

Identifying Parts of a Parabola

Identifying Parts of a Parabola

For the figure, identify the axis of symmetry, the vertex, the minimum or maximum value and explain why it is a min or a max value, identify both points and their mirror image about the AOS.

The axis of symmetry is x=-\frac{3}{2}=-1.5. Since the parabola is facing up the y-coordinate of the vertex represent a minimum value. The vertex is ( \frac{3}{2}, \frac{5}{2} ). The minimum value is \frac{5}{2}.
Point A is ( -\frac{5}{2}, \frac{9}{2} )= (-2.5,4.5) and its corresponding point would be ( -\frac{1}{2}, \frac{9}{2} )= (-0.5, 4.5).
Point B is (0, 7) and its corresponding point would be (-3, 7).

Return to the original post: click here.

Subsets of Real Numbers

Subsets of Real Numbers

Natural Numbers -are used for counting and the set does not include zero.

1, 2, 3, 4, 5, ...

Whole Numbers – include the set of natural numbers and zero.

0, 1, 2, 3, 4, 5, ...

Integers – includes  zero, the set of natural numbers, commonly called the positive integers and the opposite of the natural numbers.

... -2, -1, 0, 1, 2 ...

Rational Numbers – include any number that can be represented by the quotient of two integers. In that quotient the denominator must not be zero.

Given that a and b are integers, then \frac{a}{b} where b \neq 0.

There are all kinds of numbers that fit this description. Any combination of the fractions with the subsets above fit the definition of a rational number.

It is good to remember that all numbers that are termination decimals such as 0.25 =\frac{25}{100}= \frac{1}{4} and 0.4 =\frac{4}{10}= \frac{2}{5}.

Let’s not for get those decimals that have a pattern that can be identified as repeating. Here are just two examples: 0.\overline{6} = \frac{2}{3} and 0.\overline{4} = \frac{4}{9}.

Irrational Numbers\pi, \sqrt{2}, \sqrt{5}, \sqrt{\frac{6}{7}}, 1.021502160217...

These are just a few of the examples of irrational numbers. If a number does not fit in as one of the other numbers, it is irrational. There is another set of numbers to, but that is another topic for another day.

4.4 CPCTC Using Corresponding Parts of Congruent Triangles are Congruent

Rooted in the definition of congruent triangles, the acronym CPCTC stands for Corresponding Part of Congruent Triangles are Congruent. Basically, you have to prove the triangles congruent with what is given and once the triangle congruence is established the remaining corresponding parts are congruent.

Anytime you are required to prove corresponding parts of congruent triangles congruent, you will be doing a triangle proof, meaning that you will need to prove the triangles congruent with ASA, AAS, SAS, SSS, or HL. The examples are solved with AAS and SAS to prove the other part of the triangle congruent using CPCTC.

Example 1 Using CPCTC to Prove Two Segments Congruent

CPCTCIt is given that angle BAE is congruent to angles BCD and that segment AE is congruent to segment CD. With the given marked on the diagram you should recognize that there is one pair of corresponding angles and one pair of corresponding segments congruent. That leaves you with establishing ASA, AAS, or SAS in the two triangles.

Notice, triangles BAE and BCD share the vertex B. By the reflexive property of congruence, angle B is congruent to itself. Now triangles BAE and BCD are congruent because of AAS.

Since segment AB and CB are corresponding part of two triangles, they are congruent by CPCTC.

Example 2 Using CPCTC to Prove Two Segments Congruent

I have not yet used this in my blog, it is a modified two column proof. There is still the statements and reasons, but the reasons are included in a set of parenthesis. The statements are numbered starting with 1.


1.            segment DA is congruent to segment DC (Given)

2.            angle BDA is congruent to angle BDC (Given)

3.            segment BD is congruent to segment BD (reflexive property of congruence)

4.            triangle ABD is congruent to triangle CBD (SAS)

5.            angle ABD is congruent to angle CBD (CPCTC)

As you can see, using CPCTC is just a matter of completing a triangle congruence proof first.

Until next time,

Mr. Pi

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